I
RAYMOND CARTIER,’ DANIEL PINDZOLA, and PAUL F. BRUINS Polytechnic Institute of Brooklyn, Brooklyn, N. Y.
A Correlation f o r .
..
Particle Integration Rate in Crystal Growth Direct measurements of the growth rates of citric and itaconic acid crystals in aqueous solutions were made, using an accurately controlled flow system. Utilizing these data, a correlation for the particle integration rate of crystal growth was developed, based upon the theory and equation of Amelinckx
THE
resistance to the growth of a crystal in a solution can be considered as composed of two resistances in series: resistance to the mass transfer of material from the bulk solution to the crystal surface, and resistance to the integration of this material with the crystal. The nature of the second resistance has been the subject of speculation for a considerable time. Early crystal growth theories entirely neglected a particle integration reaction. It was believed that growth was essentially a diffusional process. As information on crystal growth and the solid state accumulated, more refined theories were put forth. All of these recognized the existence of a particle integration step, but few presented practical mathematical relationships for particle integration rate. A relationship for particle integration rate should be in convenient form, so that it can be experimentally verified and easily used. In this study of particle integration rate, linear growth velocities of faces on itaconic and citric acids were measured. Data were taken under conditions of high solution velocity relative to the crystal, where the effect of mass transfer on growth rate was negligible. Therefore, growth rate was equal to particle integration rate. A correlation for the particle integration rate of homopolar-type crystals were developed based upon the theory and equation of Amelinckx (7). This correlation fitted the experimental data very well. It substantially reduces the amount of data required to obtain an expression for particle integration rate. The correlation is easily applied to crystallizations in which the mass transfer resistance can be eliminated by such means as increasing the agitation rate. Application is also possible where the particle integration resistance normally is considerably higher than the mass transfer resistance, hence entirely controlling. Sucrose is an example of a Present address, Procter & Gamble Co.. Cincinnati, Ohio.
A generalized correlation, developed and experimentally verified by two materials, represents the particle integration rate of homopolar-type crystals. The correlation considers the fundamental as well as solution properties of the crystalline material. It is represented by the equation : G = K [ ~ P ~ ( C- ; C h - I ] . Over short ranges of temperature K varies as T3i2, while p a varies as the reciprocal of absolute temperature. Knowledge of these temperature relationships for these two terms in the particle integration rate equation enables establishment of this equation from a minimum of data: theoretically, two values of particle integration rate at two temperatures. i\ versatile method for measuring crystal growth rate permits continuous microscopic observation and repeated linear measurements of a crystal without mechanically disturbing the growth process. The variables of concentration, supersaturation, and solution velocity can be accurately controlled. The method may also be used for accurately obtaining solubility data.
system that approaches this latter condition Van Hook (73) reported that the growth rate of sucrose was relatively insensitive to variation of the agitation rate, as well as to the effect of viscosityincreasing additives. This indicates that particle integration is the major resistance in the growth of sucrose. An area where the correlation should be especially useful is in crystallizations where large single crystals are grown for their special physical properties. In such crystallizations, growth rate is a critically important process variable
Significant Literature Proposed growth rate equation based on diffusional aspect only Concept of a surface reaction introduced Experimentally confirmed existence of a surface reaction Postulated growth by repeated deposition of layers initiated by two-dimensional nuclei Screw dislocation theory of growth Effects of solution velocity on growth rate Equation for particle integration rate from a statistical analysis
(9)
(2,
4.
19)
(8) (6, 1I .
[4)
(5)
(6,$1 (1)
Development of Correlation
Amelinckx (7) developed an equation for particle integration rate from a statistical determination of the rates of particle attachment and detachment at a crystal face. Particle integration rate is the difference between these two rates. His final equation was : G =
(c, - c,)
(a
- urn vOnhile-U*/kT
) (1 1
where (C, - C,) represents supersaturation in gram; per square centimeter per second; 8 , the average linear velocity of a crystal particle in solution; m , the mass per individual crystal particle; v O , the average frequency of the lattice vibration; nhkl, the reticular density of the crystal face; t, the base of natural logarithms; U,, the attachment energy of a particle in a saturated solution; k , Boltzmann’s constant; and T , the absolute temperature. u represents a collection of terms, which have been grouped together for convenience. Its value is:
Amelinckx’s equation was not in a form which readily allowed experimental verification, especially because of the difficultyin estimating several terms in it. Therefore, a modification was sought VOL. 51, NO. 11
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NOVEMBER 1959
1409
which would enable a linear correlation of experimental data and would make it unnecessary to know many of the difficultly evaluated terms. The terms which are not known in most cases, and which are not easily estimated, are: ii, a, V O , nhk13 and U,. Because these variables are predominantly functions of temperature over moderate ranges of supersaturation, an attempt was made to group them into a term that could be easily expressed as a temperature function. A modification was found which satisfactorily correlated the particle integration rate data of citric and itaconic acids. G
= K[eu\Ci-Cd
- 11
-OVERFLOW
I
COOLING
I
I
I
I
I‘
BYPASS LINE
I
UNIT
ISOLUTION I
!
I,
j.
ICROUETER
TO S E W E R
HEATING UNITS
I
1
I
(3)
--
where CRYSTAL
A recirculating solution system was used to measure rate of crystal growth
K was named “the particle integration factor.” The logarithmic form of Equation 3, in which concentration units have been converted to weight per cent, is : In (G
+ K ) = In K +
pu
(C;
- Ci)
(5)
where p represents solution density. A single value for p can be used for conditions at saturation and at low values of supersaturation. Equation 5 represents a convenient means of correlating particle integration rate data. O n plotting (G K) against (C; - Ci) on semilog paper, a straight line results. Its slope is equal to pa. A trial and error procedure for K must be used. However, the correct value is easily ascertained from a check of the intercept, which must equal In K . A more accurate means of correlation is afforded if Equation 3 is analytically solved. It was found that K could be expressed as the absolute temperature raised to the 3/2 power: and that p a could be expressed as the reciprocal of absolute temperature. Therefore, the only data
+
H O U S I N G UNIT
required to apply the correlation are particle integration rate and solubility data. As the temperature relationships for K and pu are known, a theoretical minimum of only two values of particle integration rate is needed at two temperatures to establish the relationship for particle integration rate. Development of the Equation. The right-hand side of Amelinckx’s equation (Equation 1) was multiplied by u,’u, giving : G
=
.(c,
-
6 c,) (G -m
YOnhdi
e-Ua/kT
)
(la)
- 11
G = 0
(G-
m
Yonhk2
e-Li./kT)
(IC)
The second grouping of terms in Equation I C was given the symbol K . Specifically,
To develop the temperature relationship for K , 6 was expressed in the form derived by Maxwell. However, an attenuation factor, u, was applied to account for the differences between perfect gases and solutions. Thus:
Expanding e‘(C1-C,) by means of a McLaurin series gives: ea(c1-C.)
= 1
+ u(C1 - C,)
(lb)
This expansion is valid for low values of the exponent. Solving this expansion for u(C1 - C,) and substituting this value in Equation l a , gives:
r
From the experimental results it was seen that u increased just slightly with temperature. Therefore, because a could also be expected to increase slightly with temperature, a direct
CONCENTRATION
I N INT %
C I T R I C bClD 4NHYDROUS
lT4CONlC
SOLUTION
42 3
VELOCITY I N FE~?/SECOND
30
Figure 1. Growth rate becomes independent of solution velocity as velocity is increased
1 4 10
INDUSTRIAL AND ENGINEERING CHEMISTRY
Figure 2. Growth rate data of the ( 1 1 1) face of citric acid monohydrate
CRYSTAL GROWTH proportionality between these two t e r m could be assumed over short ranges of temperature. This allows the first term in K to be written :
where a represents a constant. The second grouping of terms in Knamely, mvg n h k l e-UaIkT-has a relatively constant value over a short temperature range, since an increase in temperature decreases e - Us/k", while v o increases. Therefore, the temperature relationship for K can now be written as: K =
where
- p
CuT3'2
regulator. As the solution was returned to the reservoir, it passed through a rotameter and by a thermometer, which were used for observing solution flow rate. A bypass line around the crystal housing permitted placement of a crystal within the housing. A bulb syringe was attached to the crystal housing, so that solution could be displaced from the housing prior to removing the groundglass stopper. City water was used as the cooling medium in the cooling exchangers. I t was first fed to a 3-gallon insulated stand tank located 9 feet above the apparatus.
The flow of cooling water from the stand tank to the exchangers was regulated by means of a needle valve. A constant head was automatically maintained in the stand tank by having water continuously run off through a vent at the side of the stand tank. The heat exchangers were supplied with hot water pumped from a thermostatically controlled 5gallon reservoir containing an 1800-watt immersion heater. Growth rate data were taken at various solution concentrations. Before growth rate measurements were made, a freshly introduced crystal was partially dissolved. then regrown. in order to remove stresses
(Igl
p is a constant defined by: @ = m
uOnhk, e - b ' d k T
\I3 6 4
(Ih)
Equipment and Experimental Procedure
The apparatus used to measure crystal growth rate consisted of a recirculating solution system, provided with a section where a fixed crystal could be microscopically observed while the linear growth of one of its faces was measured. T h e variables studied were concentration and supersaturation. Solution velocity was not a variable in the study. However, it was possible to vary and control it in the apparatus used. Constant conditions could be maintained within the apparatus, so that temperature variation was within 0.1 " C. The solution flow path was as follows: Unsaturated solution from the 21,'2gallon constant temperature reservoir was pumped to cooling exchangers, where it was cooled to the desired degree of supersaturation by means of cold water. The supersaturated solution then passed a calibrated 0" to 50" C. glass thermometer, which preceded the crystal housing. The crystal housing contained a crystal 0.5 to 1.0 mm. in length attached to the tip of a 1-mm. tungsten wire by means of a minute amount of Canadian balsam cement. The wire was embedded in a glass rod, which was attached to a ground-glass stopper, Linear measurements of the growth of a particular crystal face were made at a magnification of 1OOX by an 4 . 0. Spencer screw micrometer eyepiece. T h e solution passed from the crystal housing to heat exchangers, where it was heated by means of hot water to within 1 to 2' of the temperature in the solution reservoir. Thus a substantial quantity of the heat removed in the cooling exchangers was returned to the solution. Constant temperature was maintained in the solution reservoir by a 600-watt glass immersion heater activated by a mercury expansion thermo-
Figure 3.
Growth rate data of the (001) face of itaconic acid
G R O W T H RATE IN MM /MI N.
18
20
22
TEKPERAYURE % oc.
30
32
34
Figure 4. Growth rate as a function of supersaturation a t different temperatures was obtained from a cross plot such as this one for itaconic acid VOL. 51,
NO. 11
NOVEMBER 1959
141 1
and other imperfections. Growth rate could then be measured at various degrees of supersaturation by adjusting the needle valve that controlled the flow of cooling water. At each concentration studied, measurements were first made at the same supersaturation, but at different solution flow rates, to ensure obtaining data that were independent of the effects of mass transfer. Figure 1 shows the type of results obtained as flow rate was varied a t fixed supersaturation. I n the case of itaconic acid, the critical flow rate lies below a solution velocity of 0.6 foot per second. Solubility data were obtained by microscopically observing the edges of a crystal as the degree of supersaturation was gradually reduced to zero. These edges became rounded a t the saturation temperature. The ability to determine solubility data accurately was a particularly useful feature of the apparatus, as such data were necessary in this study. A Vanton pump having a Buna-K block and Hypalon Flexi-liner was used to circulate the solution. I t was driven by a variable-speed '/s-hp. direct current motor. All flow lines and accessories
in contact with the solution were made of borosilicate glass, except the line connections, which were of neoprene. The solutions used were made using distilled water and anhydrous citric acid, U.S.P. grade, and refined grade itaconic acid (Pfizer). Solution concentrations for both materials were determined by titration with 0.1,V sodium hydroxide, using phenolphthalein as the indicator.
would have interfered with the growth process. The solubility data experimentally obtained were correlated by means of an Othmer plot (70))in which the logarithm of solute mole fraction was plotted against the logarithm of the vapor pressure of water at the same temperature. The equations representing the correlated solubility data are : (Citric acid monohydrate)
Experimental Results
log x = 0.3484 log
The growth rate data obtained for the (111) face of citric acid monohydrate and the (001) face of itaconic acid are presented in Figures 2 and 3, respectively. These figures represent the data as they were recorded. They illustrate that growth rate, where particle integration is the controlling process, increases rapidly as the solution temperature is reduced below the saturation temperature-i.e., as supersaturation is increased. If an attempt had been made to obtain data at temperatures 2' to 3 C. below the minimum temperatures indicated in the diagrams, a high rate of nucleation would have occurred, which
P' -
1.3568
(6)
(Itaconic acid) log x = 0.7510 log P' - 2.0061
(7)
in which x is the solubility expressed as the mole fraction in water, and P' is the vapor pressure of water in millimeters of mercury a t the corresponding temperature. Values of the saturation concentrations, Ci, were calculated from Equations 6 and 7 at the desired temperatures. To calculate growth rate and supersaturation values conveniently at these temperatures, cross plots of Figure 2 and 3 were made, in which concentration, C;, was plotted against temperature
4 Figure 5. Correlated growth rate data for the ( 1 1 1 ) face of citric acid monohydrate Figure 6. Correlated growth rate data for the (001) face of itaconic acid
4
100
h
Y
+ (3 v
c3
0 _I
9
4v/ 2
14 12
INDUSTRIAL AND ENGINEERING CHEMISTRY
c
CRYSTAL GROWTH
J
t
c
/
!
/
31
-’
/
L
3 5 1
/’
t-
t
1
/
3 0
//
1
I
/
t
x 2 9
c
y
-
J
/’
J
1
I
2.1 5000
5200
I
I
3/2
5400
I
I 5600
Figure 7. The particle integration factor for the ( 1 1 1 ) face of citric acid monohydrate i s a linear function of T 3 I 2
with growth rate, G, as parameter. Figure 4 illustrates the cross plot of Figure 3 for the data of itaconic acid. The growth rate data of citric acid monohydrate and itaconic acid were correlated by means of Equations 3 and 5 . The method of averages was applied in the algebraic correlation of these data. The graphical correlation of these data is presented in Figures 5 and 6. The values of K and p a obtained are given in the table below. The linear temperature relationship for K described by Equation l g is verified in Figures 7 and 8. The equations representing these curves are :
Figure 8. The particle integration factor for the (001) face of itaconic acid is a linear function of T312
Itaconic acid K ~ . =~ (1.212 , 773’2 - 3260) (10-9) (9) I t was found that the term pcr could be represented as a function of the reciprocal of absolute temperature. Figures 9
Citric acid monohydrate
KC.A.M.
=
(4.502 T3’2- 20:820)
(8)
0.90
1
1
0.8 6
0.84
b Figure 9.
Temp., O
c.
PU
K X IOe, G . / Sq.Cm.-See.
Itaconic Acid
20.00 21.00 22.00 24.00 26.00 28.00 30.00
0.708 0.713 0.720 0.724 0.724 0.729 0.730
2.740 2.852 2.920 2.936 2.990 3.042 3.151
Citric Acid Monohydrate 23.00 26.00 29.00 32.00 35.00 38.00
0.757 0.761 0.799 0.820 0.858 0.880
2.101 2.400 2.640 3.025 3.463 3.782
The term p a of citric acid monohydrate varies linearly as 1/T
Y O . 82
0.8 0
0.7 S
1
i
-
i 1
1
1
I
1
VOL. 51, NO. 1 1
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Nomenclature
attenuation factor solution concentration, weight % solution concentration, grams/cc. base of natural logarithms, 2.71828,. , G = growth rate specifically where particle integration is the controlling process, grams/sq. cm.second I; = particle integration factor, grams/ sq. cm.-second k = Boltzmann’s constant, 1.3805 X 10-’6 ergs/’ K.-molecule m = mass of individual crystal particle, grams n h ~ i . = reticular density of hX1 plane, number of molecules/sq. cm. P‘ = vapor pressure of water, mm. of mercury T = absolute temperature. ’. L’ = attachment energy of crystallizing particle, cal./particle D = average velocity of crystal particles in solution. cm./second i = mole fraction dissolved in water = average frequency of the lattice YO vibration. vibrationsjsecond x = Di. 3.1416. . p = solution density, grams/cc. ( bu/bC)c ce g = kT ,unitless a
C’ C e
Figure 10.
The term
p a of itaconic acid
varies linearly as 1 / T
= = = =
I
3.3
VTx I o 3
SCBSCRIPTS
represented by Equation 3, the equations are:
and 10 represent this empirical temperature relationship for pa. The equations for these curves are : PflC..k.M.
= -
8p+
-
3.7070
( 1 11 ) face of citric acid monohydrate
G = (1.212 T3’2- 3260)(10-9) X
-
7+
1.3977)(C;
-
C:
1- 1]
(13)
,TUBING T O B U L B S Y R I N G E GROUND-GLASS
TUNGSTEN
FlTlNG
If
/ ,
0
WIRE
DIAMETER IMM.
‘FLOW
DIRECTION
&i
OF SOLUTION
=
1
= conditions of supersaturated
Acknowledgment
literature Cited
(001) face of itaconic acid
[e(
conditions of saturated solution solution near crystal face
s
The authors thank Chas. Pfizer & Co. for generous financial support of this work.
G = (4.502 T’” - 20,820)(10-9) X
(10)
The final expressions for the growth rates of the particular faces of citric acid monohydrate and itaconic acid studied can now be written. Using the form
>
,
i
I
(1) Amelinckx, S., J . chim. phys. 47, 213 (1950). ( 2 ) Berthoud, A., Ibid.,10,624 (1912). (3) Frank, F. D., Discussions Faraday SOC. 5 , 4 8 (1949). (4) Friedel, G., Bull. soc. franG. mineral. 48, 12 (1925). ( 5 ) Hixson, A. W., Knox, K. L., IND. E N G .CHEM.43, 2144 (19511. ( 6 ) Kossel, W., Nachr. Ges. Wiss., Gottinpen, Math. physik. Klasse 1927, p. 135. ( 7 ) McCabe, W. L., Stevens, R. P., Chem. Eng. Progr. 47, 168 (1951). (8) Marc, R., Z . physik. Chem. 79, 71 (1912). ( 9 ) Nernst, W., Ibid.,47, 52 (1904). (10) Othmer, D. F., Thakar, M. S., IND. ENG.CHEM.44, 1654 (1952). jtranski, I . N., Z . physik. Chem. 136, (1928). Jaleton, J. J. P., Z . Krist. 60, 1 (1924). (13) Van Hook, A., Frulla, F., IND.ENG. CHEM.44, 1305 (1952). (14) Volmer, M., Z. physik. Chem. 102, 267 (1922).
RECEIVED January 2, 1959 ACCEPTEDJuly 1, 1959
A crystal housing made possible microscopic observation of a fixed crystal 14 1 4
INDUSTRIAL AND ENGINEERING CHEMISTRY
Division of Carbohydrate Chemistry, Symposium on Crystal Growth from Solution, 134th Meeting, ACS, Chicago, Ill., September 1958.
